PLANE WAVES OF SOUND 167 



provided T = ZMc/tca = ^Mjp^c, n' z = n*-l/r*. i ..... (18) 



When nr is large the effect on the period may be neglected. 

 The condition for this is that 2M* must be large compared 

 with p o>X/27r, where \ is the wave-length. The inertia of the 

 piston must therefore be great compared with that of the air 

 contained in a length X/2-n- of the tube. The same law of 

 decay would be given also by the indirect method explained 

 in 12. 



We have seen in (13) that the rate of propagation of energy 

 across unit area of wave-front in a progressive system of waves 

 of simple-harmonic type is Jp ri 2 a a c, or J/^oC 3 ^ 2 , if s l denote the 

 maximum condensation. The result was obtained for plane 

 waves, but will hold for all kinds of wave at a sufficient distance 

 from the source. Consequently if W denote the total emission 

 of sonorous energy per second from a source near the ground, 

 the value of s a , at a distance r, will be given by the relation 



W =i/3 c 3 5 1 2 x 27rr 2 = 7rp c 3 r 2 5 1 2 .......... (19) 



This formula was applied by Lord Rayleigh to estimate the 

 limit of audibility of a sound of given pitch. The value of W, 

 as inferred from the power spent in actuating the source 

 (a whistle), is the product of the current into the pressure, and if 

 r be the distance at which the sound is just audible, the formula 

 will give a value of s lf which is necessarily, however, greater 

 than the true limit, since the value of W is too high, not all 

 the energy being spent in sound. In this way it was ascer- 

 tained that sounds could be heard in which Sj_ was certainly less 

 than 4 x 10~ 8 . The corresponding amplitude as deduced from 

 the formula ncu cs^ was 8 x 10~ 8 cm. By an independent 

 method, in which the above source of uncertainty was avoided, 

 the limit of audibility was fixed at about s l = 6 x 10" 9 . Subse- 

 quent experiments by Wien (1903) and Rayleigh f indicate an 

 increase of sensitiveness with rise of pitch, for tones near the 

 middle of the ordinary musical scale. 



* The factor 2 would disappear if the piston were supposed to generate waves 

 on both sides. 



t Phil. Mag. (6), vol. xiv. (1907). 



